This thesis investigates aspects of field theories and soliton solutions with nontrivial topology. In particular we explore the following effective models: a limited sector of the scalar Electroweak theory called extended Abelian Higgs model, and a classical mechanics model derived from the low energy SU(2) Yang-Mills theory.
The extended Abelian Higgs model applied on two-component plasma of charged particles is studied numerically. We find evidence that the model admits straight twisted line vortices. The result is described by an energy function that acquires a minimum value for a non-trivial twist. In addition to the twisted line vortices the result also suggests that stable torus shaped solitons are solutions of the theory.
Furthermore we construct a classical mechanics model exhibiting some of the key properties of the low-energy Yang-Mills theory. The dynamics of the model is studied numerically. We find that its classical equations of motion support stable periodic orbits. In a three dimensional projection these trajectories are self-linked in a topologically non-trivial manner suggesting the existence of knotted configurations in low energy SU(2) Yang-Mills theory.
We calculate the one-loop effective action for the Abelian Higgs model with extended Higgs sector. The resulting first order quantum corrected model shows close resemblance to a modified model where texture stabilizing term has been added to the system. In the limit where the gauge field can be entirely expressed by the scalar fields, the both models become identical suggesting that the theories are closely connected. This implies that quantum corrections have stabilising effect on the soliton solutions.
These studies have contributed to a better understanding of the dynamics of non-linear low energy systems, and brought us a step closer to exploring full scale physically realistic models.